tailieunhanh - Báo cáo hóa học: "MULTIPLE PERIODIC SOLUTIONS FOR A DISCRETE TIME MODEL OF PLANKTON ALLELOPATHY"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: MULTIPLE PERIODIC SOLUTIONS FOR A DISCRETE TIME MODEL OF PLANKTON ALLELOPATHY | MULTIPLE PERIODIC SOLUTIONS FOR A DISCRETE TIME MODEL OF PLANKTON ALLELOPATHY JIANBAO ZHANG AND HUI FANG Received 19 May 2005 Revised 25 September 2005 Accepted 27 September 2005 We study a discrete time model of the growth of two species of plankton with competitive and allelopathic effects on each other N1 k 1 N1 k exp r1 k - a11 k N1 k -a12 k N2 k - b1 k N1 k N2 k N2 k 1 N2 k exp r2 k - a21 k N1 k - a22 k N2 k - b2 k N1 k N2 k . A set of sufficient conditions is obtained for the existence of multiple positive periodic solutions for this model. The approach is based on Mawhin s continuation theorem of coincidence degree theory as well as some a priori estimates. Some new results are obtained. Copyright 2006 J. Zhang and H. Fang. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Many researchers have noted that the increased population of one species of phytoplankton might affect the growth of one or several other species by the production of allelopathic toxins or stimulators influencing bloom pulses and seasonal succession. The study of allelopathic interactions in the phytoplanktonic world has become an important subject in aquatic ecology. For detailed studies we refer to 1 2 7 9-11 13 and references cited therein. Maynard-Smith 9 and Chattopadhyay 2 proposed the following two species Lotka-Volterra competition system which describes the changes of size and density of phytoplankton N1 n - a11N1 t - aMt - b1N1 t N2 dt N1 r2 - a21N1 t - a22N2 t - 2N1 t N2 t dt where b1 and b2 are the rates of toxic inhibition of the first species by the second and vice versa respectively. Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 90479 Pages 1-14 DOI ADE 2006 90479 2 Periodic solutions for a discrete plankton model Naturally more realistic models .